The relationship between the twins in general counts. Sure, they pester each other a lot, but not five minutes go by when it's not clear they're the best of friends. And after so many Disney Channel shows where any siblings in the show are almost always at each others' necks with the sweet moments few and far between, the only exception being another animated show taking place in the summer, it's a real breath of fresh air.
The little interactions between Dipper and Mabel (grabbing her hand and leading her away from danger, pouring a soda for her at the table in Double Dipper, the high-fives, the affectionate punches, etc.) really enforce that they truly are the best of friends. It's sweet.
Some of Alex Hirsch'stweets make it clear how much he and his sister's childhood influenced the show. You can tell he put his heart into it!
Lance Bass appearing in the episode "Boyz Crazy" was a gift from Alex to Ariel, who was obsessed with the singer when she was younger.
Wendy and Dipper's interactions in general are very sweet. Wendy is such a nice character in general - she hangs with friends her age and is a rowdy teen, alright, but she is never condescending to the kids and treats both Mabel and Dipper as equals, goofing around with them when, really, she doesn't have to at all. That, and Dipper's crush on Wendy is in general kind of adorable and well portrayed, as also seen in "Double Dipper".
Wendy and Dipper "ganging up" on the video game ref in the beginning of Fight Fighters was oddly adorable before Robbie showed up, anyway.
Grunkle Stan's Pet the Dog moment. Considering what we've seen of his character so far, letting Dipper and Mabel each pick something from the Mystery Shack gift shop is a pretty big deal.
"This journal told me there was no one in Gravity Falls I could trust. But when you battle a hundred gnomes side by side with someone, you realize that they've probably always got your back."
The theme of trust between Dipper and Mabel that sets up their teamwork for the rest of the show is done really well, and also gives us a lot of really adorable sweet sibling moments. Considering this is the first episode, there are sure to be more.
The Legend of the Gobblewonker
After abandoning Grunkle Stan to go chase a monster that turned out to be just a very tech savvy hillbilly who explained that he wanted some attention from his family. Dipper and Mabel (and Soos) decide to go back and spend time with each other like poor Stan originally planned to.
Dipper: Well, we spent all day trying to find a legendary dinosaur...
Mabel: But we realized the only dinosaur we wanna hang out with is right here.
Grunkle Stan affectionately noogie-ing Dipper and Mabel at the end.
The Hand That Rocks the Mabel
Mabel saving the soon-to-be-eaten lobster and keeps him as a pet.
Dipper helping Mabel out of Sweater Town, followed by him offering to break the news about her wanting to break up with Gideon for her so she wouldn't have to. That was seriously nice of him, even if it had unintended consequences.
The reason why Mabel is so enthusiastic about befriending Gideon at first is that she likes having someone to do girly things with, since Soos and Dipper do "boy things" all the time. How does the episode end? With Mabel bedazzling Soos.
Grunkle Stan, Dipper and Mabel bonding over their shared dislike for Lil' Gideon.
Wendy lying to her friends about how Dipper defeated the ghosts in so that he could keep whatever dignity he had left, even though she clearly loved his dance.
Dipper vs. Manliness
after Dipper tells his story about the Manotaurs and the Multi-Bear, Grunkle Stan points out how doing the right thing in spite of the pressure is rather manly. Let me repeat, Grunkle Stan said this.
Dipper choosing not to kill the Multi-Bear, despite all the pressure to do so, simply because it wasn't the right thing to do.
It's a brief moment, but in the diner, Manly Dan can be seen with his daughter Wendy, eating pancakes with her. It lead to the interpretation that this is how Manly Dan bonds with his daughter: over breakfast. Even if her brothers weren't around, it was nice to see Manly Dan interact with Wendy.
Mabel's new friends staying with her even though she lost the dance contest to Pacifica.
Candy: We may not have as many friends as Pacifica, but we have each other, and that is just as good, I think.
Also in the episode, Grunkle Stan cheers for Mabel and wanted her to win the contest. The other contestant wasn't the nicest regardless, but it goes to show that Grunkle Stan does care about Mabel.
Again, Dipper and Wendy have a fairly sweet moment. It's kind of downplayed, but their casual conversation when Dipper runs into her near the bathroom is just... nice.
Then there's the final scene with Dipper and Tyrone just talking together about what they've learned about not getting in their own way.
And Tyrone giving Dipper extra encouragement to go after Wendy as he melts away.
Earlier, Tyrone trying to feed Paper Jam Dipper. It's oddly heartwarming and funny and kind of creepy.
Dipper comforting Mabel after Pacifica treats her like crap, followed by a pep talk.
Dipper exposing the truth about Pacifica's family to her after her harsh treatment of Mabel, despite the fact that Mabel no longer cared at that point; he loves his sister enough to get revenge for her that she didn't even want.
The Time Traveler's Pig
Dipper giving up his hard fought time travel assisted friendship boost with Wendy, so Mabel can keep her new pig, Waddles. The thank-you hug she gave him afterwards was nothing short of adorable.
"That'll do, pig. That'll do."
Hell, even the intro of Gravity Falls introducing Mabel cuddling a pig is this.
"Did you hear that? She called me one of her boys!"
Mabel helping Grunkle Stan get over his fear of heights was both hilarious and adorable.
Dipper sticking a bandaid on Mabel's skinned elbow in "Summerween".
In a somewhat creepy way, the Summerween Trickster being able to die happy in the knowledge that someone enjoyed eating him.
3 year old Dipper & Mabel dressed as kittens. D'awww...
When Mabel is sad that even after all that they didn't get any candy, Stan reveals the two bags left behind by those two kids from earlier.
The LolPigs segment during the credits gives a really warm fuzzy feeling to watch, especially since this was one of the freakiest and most intense episodes in the entire show.
Grunkle Stan trusts Mabel with the shack for 72 hours in "Boss Mabel". He does it to prove a point. However, its not often that Grunkle Stan acts are fueled by principle instead of money.
Dipper admits he misses spending time with Mabel and feels like he's getting left behind when she goes to hang out with her friends. That, plus the fact that they move back into the same room by the end of the episode, just shows how close they are and that they value each others' company more than anyone else's (which is arguably just about the sweetest thing in the world).
Dipper: Hey, um... Do you want to have a sleepover?
Mabel telling Grenda that she's too beautiful to need make-up, although probably intended as a joke given Grenda's "butch" appearance, is very sweet and genuine coming from Mabel.
Most of Stan's interactions with Mabel in Dipper's body.
After realizing he doesn't need the spare room after all, Dipper gives it to Soos as a new employee lounge, to replace the cramped, hazardous "break room" in the crawlspace.
One of the first things Soos does after regaining his proper body is scratch the itch Waddles can't reach on the back of his head.
Dipper exposes Robby's song of having subliminal messages. Grunkle Stan states that he did the right thing and that until Wendy gets her life back together after dumping Robbie, the two of them could always go bowling together. There's also the scene where they give each other high-fives and the fact that Grunkle Stan knows what it's like for Dipper to lose someone he loves to music to be there to support him.
Somewhat Fridge-Heartwarming, but the fact that Stan cares enough about Wendy (his lazy, sarcastic employee who pelted him with water balloons for no reason two episodes prior) to actually believe Dipper's shenanigans and take a dangerous, off-road route to save her.
In the end, Mabel decides to let Sev'Ral Timez go and be free, even though she's still torn about how every guy she's met leaving her.
On a meta level: Series creator Alex Hirsch's sister Ariel had a huge crush on *NSYNC's Lance Bass as a kid, so Alex cast Bass as a guest in this episode and invited Ariel to come meet him and get memorabilia signed. He also gave her a cameo in the episode. The code at the end of the episode reads HAPPY NOW, ARIEL?
The Land Before Swine
Stan and Mabel making up after their fight, culminating in the two of them falling asleep together covered in tree sap.
Dipper and Soos also making up after their fight with their fist bump.
The reason Stan acts so cold and bossy towards Dipper, and constantly making him do chores? He sees how weak he was when he was a young man in Dipper. Just like what his father did to him, Stan's toughening him up so he can become strong and proud.
He specifically says he does it so Dipper can fight back whatever the world throws at him. Considering all the danger in Gravity Falls (which Stan might already know about), it makes it even more heartwarming.
A minor one for the same episode comes from Bill Cipher. He may have pulled out that deer's teeth, but it was good to see that he actually bothered to restore the deer's teeth after he was done making his deal with Gideon.
Also, take a look at the wall by the door in Stan's office during Gideon's first break-in attempt. There's a picture of a smiling Soos with a plaque labeled "Employee of the Month."
Grunkle Stan being legitimately upset when he has to send Dipper and Mabel home because he can't take care of them, anymore. Even for all of the teasing and such, he really does love them.
At the end when Stan is considered a hero for exposing Gideon as a fraud to the whole town.
While affectionately noogie-ing Dipper and carrying Mabel on his shoulder, no less!
The montage at the Mystery Shack showed they were together again and that the Shack was finally theirs.
Special mention goes to Dipper and Mabel telling Stan about the journal, Soos water gunning them, and Mabel telling Dipper he was still a hero without the journal.
A subtle one is they kept calling the Mystery Shack their home.
Stan even has a framed photo of both Dipper and Mabel in the secret facility behind the candy machine.
Dipper and Wendy smiling at each other while they repair the Mystery Shack roof.
Dipper and Mabel desperately trying to hold on to each other as Gideon-bot picks them up.
After being picked up, Dipper screams at Gideon to let Mabel go, not really seeming to care that he was in the same predicament himself.
Grunkle Stan giving Dipper and Mabel most of the credit for exposing Gideon. The Pines family being town heroes in general is this.
Him going all Papa Wolf to protect the kids from the zombies and revealing he always knew the town was crazy, but lied to protect the kids.
In a small one, Wendy playing look-out to help Dipper.
Most importantly, the trio karaoke song they sang as a family and defeat the zombies.
Dipper: I'm sorry about this, guys. I totally ruined everything. Mabel: Dipper, are you kidding me? I got to sing karaoke with my two favorite people in the world! No party could ever top that.
What Dipper says as he's about to be eaten by a zombie:
The glowing praise she gives Dipper about how much their friendship means to her:
Wendy: Lemme tell you something: this summer was SUPER boring until you showed up. I have more fun with you than, like, practically anybody else; and if you ever stopped being my friend, I would, like, throw myself into the Bottomless Pit!
The Golf War
Grunkle Stan giving Mabel a "U Da Best!" sticker before her mini-golf duel with Pacifica. It ends up causing more trouble than its worth, sure, but he didn't know that!
Stan is generally a big sweetheart the entire episode, to both the twins and to Soos. His Character Development is kicking in and it shows.
The ending, where Mabel offers Pacifica a ride home in Grunkle Stan's car and one of the tacos she found in the back-seat, and Pacifica seems to slowly move towards being, at the least, Vitriolic Best Buds with Mabel.
Pacifica: And Mabel, I can't believe I'm saying this, but... I had fun.
Also, in spite of calling him a "servant" while doing so, she gives a sincere compliment to Soos about his fashion update, which please him greatly.
Big Henry bravely sacrificing his life to push the ball through a deadly gas leak. In his last moments, he even looks at a drawing Polly made of the two of them together and tears up with a proud smile before he dies. Of course, the absurdity and low stakes of it all is Played for Laughs.
Mabel basically giving up her chance to impress her Crush of the Week by preventing Dipper!Bill from taking the book, destroying the set she worked so hard to build in the process.
Dipper!Bill: I mean, who would sacrifice everything they've worked for just for their dumb sibling?
Mabel: ...DIPPER WOULD.
By this same token, Mabel coming to the realization that her obsessions and boyfriend-of-the-week habits, as well as her expectation of Dipper to just roll with them, is actually very draining on Dipper and selfish of her to boot. It really goes to show how talented the writers are that they were able to take one of Mabel's negative character traits and turn out an episode based solely on that that lets her learn from the experience and become a better person.
From Mabel's Guide To Dating, Soos gives us this.
Soos: How many kids do I want to have? Seven, preferably. That's one to love for every day of the week!"
Mabel's "masterpiece" at five years old was a squiggly stick-figure drawing of Dipper.